We examine the gluon propagator in the Laplacian gauge in quenched lattice
QCD as a function of the number of colours. We observe a weak dependence on
$N_c$ over the whole momentum range. This implies an almost $N_c$-independent
gluon pole mass in units of the string tension.Comment: 5 pages, 7 figures. A study of the gluon propagator in SU(4) for
  various lattice sizes was added. Extended version to appear in Phys. Rev. 

Alexandrou, C.

de Forcrand, Ph.

Follana, E.

English

arXiv

We examine the gluon propagator in the Laplacian gauge in quenched lattice QCD as a function of the number of colours. We observe a weak dependence on $N_c$ over the whole momentum range. This implies an almost $N_c$-independent gluon pole mass in units of the string tension.We examine the gluon propagator in the Laplacian gauge in quenched lattice QCD as a function of the number of colours. We observe a weak dependence on $N_c$ over the whole momentum range. This implies an almost $N_c$-independent gluon pole mass in units of the string tension

De Forcrand, P.

CERN Document Server

The Laplacian Gauge Gluon Propagator in SU(Nc)C. AlexandrouDepartment of Physics, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, CyprusPh. de ForcrandInstitut fu¨r Theoretische Physik, ETH Ho¨nggerberg, CH-8093 Zu¨rich, SwitzerlandandCERN, Theory Division, CH-1211 Geneva 23, SwitzerlandE. FollanaDepartment of Physics, University of Cyprus, P.O. Box 20537, CY-1678 Nicosia, Cyprus(Feb. 15, 2002)We examine the gluon propagator in the Laplacian gauge in quenched lattice QCD as a functionof the number of colours. We observe a weak dependence on Nc over the whole momentum range.This implies an almost Nc-independent gluon pole mass in units of the string tension.PACS numbers: 11.15.Ha, 12.38.Gc, 12.38.Aw, 12.38.-t, 14.70.DjI. INTRODUCTIONLattice QCD provides a well suited theoretical frame-work for the study of nonperturbative quantities as afunction of the number of colours. Although experimen-tally inaccessible, QCD for Nc = 2 and Nc = 4 is in-teresting from the theoretical point of view. RecentlySU(Nc) Yang-Mills theories were investigated for vari-ous Nc by two groups [1,2] with emphasis on the eval-uation of string tensions and glueball masses and theirlarge Nc limit. It is the purpose of the present work toexamine the gluon propagator for dierent numbers ofcolours. Like in our previous work in SU(3), we evalu-ate the gluon propagator in the Laplacian gauge which,unlike the more usual Landau gauge, is free of spuriousGribov copies on the lattice. Therefore no doubt can becast on our results due to the lattice gauge xing pro-cedure. Furthermore theoretical arguments have beenpresented supporting gauge invariance of the pole massof the transverse gluon propagator [3] and it is thereforean interesting quantity to investigate as a function of Nceven in a particular gauge. Studying the dependence ofthe gluon mass on Nc has theoretical implications forconnement and chiral symmetry breaking. It has beenargued that a non-zero gluon mass is connected to vortexcondensation [4] and to the glueball mass [5]. Since theNc dependence of the glueballs has been investigated, astudy of the gluon mass can shed light on their relation.In Dyson Schwinger studies the form of the gluon propa-gator aects that of the quark propagator, which in turnprovides a determination of the quark mass and thereforeis connected to chiral symmetry breaking.In previous studies [6,7] we considered the gluon prop-agator in SU(3), and presented evidence for a pole whichsurvives the continuum limit. In this work we present re-sults in SU(2), SU(3) and SU(4) at similar couplings.We refer the reader to ref. [6] for our notation and thedetails of our approach.II. LAPLACIAN GAUGE FIXINGThe Laplacian gauge consists of rotating along a xedorientation the local color frame built from the lowest-lying eigenvectors of the covariant Laplacian. It is welldened in the continuum theory but non-perturbative.On the lattice, it has the virtue of being unambiguous(except for genuine Gribov copies arising from accidentaldegeneracy of Laplacian eigenvalues), unlike the latticeLandau gauge.In the case of SU(2), the Laplacian eigenvalues aretwofold degenerate [8] due to charge conjugation symme-try: if f is an eigenvector so is σ2f?. At each latticesite, one can construct from these two vectors a 2  2matrix which is then projected onto SU(2) and denesthe required gauge rotation.In SU(3) and SU(4), we identify the Nc−1 lowest-lyingeigenvectors of the Laplacian. From these eigenvectorswe construct at each lattice site an SU(Nc) matrix byusing a Gram-Schmidt orthogonalization, as described indetail in ref. [6].III. RESULTSIn order to make the comparison of our results inSU(2), SU(3) and SU(4) easier, we consider β val-ues which yield similar lattice spacings in units of thestring tension σ. We take β = 2.4, 6.0 and 10.9 forSU(2), SU(3) and SU(4) respectively, which give apσ =10.2634(16) [13], 0.2265(55) [9] and 0.2429(14) [1] for thethree gauge groups. We can then express our SU(Nc)results in \physical" units usingpσ = 440 MeV. The in-verse lattice spacing is 1.67(1), 1.94(5) and 1.81(1) GeVfor SU(2), SU(3) and SU(4) respectively.We rst examine SU(2). For this gauge group, severallattice studies of the gluon propagator in the Landau andColoumb gauges exist [10,11]. We present here the rststudy in the Laplacian gauge where no issues of latticeGribov copies arise. It is therefore important to have lat-tice artifacts under control. The volume dependence isinvestigated by considering the SU(2) gluon propagatorfor lattices of size 44, 84, 164, 163  64 and 324. Thequantity of interest is the transverse part, D(q2), of thegluon propagator. This is shown in Fig. 1 in lattice unitsfor the various volumes after making a cylindrical cut [12]in the momenta which eliminates the leading lattice ar-tifacts due to lack of rotational symmetry. For non-zeromomenta, we subtract the longitudinal part of the propa-gator to project out its transverse part. For zero momen-tum this subtraction is not well dened, and we keep thecomplete propagator. We include results from the veryasymmetric 16364 lattice. Even in that case, only smallnite-size eects are visible in the infrared (momenta lessthan  0.5 GeV in \physical" units).FIG. 1. The SU(2) gluon propagator in lattice units forvarious volumes.One conclusion to be drawn from Fig. 1 is that thezero momentum gluon propagator has a nite value asthe volume is increased. In fact for lattices of spatialsize greater than 0.8 fm it reaches a constant value. Thisbehaviour also holds for SU(3) [6,7]. For SU(4) we haveperformed simulations for only one lattice size since thesimilarity of the results in SU(2) and SU(3) leads us toexpect the same behaviour also in SU(4).In Fig. 2 we make a comparison of the renormalizedzero momentum gluon propagator, D(0) in \physical"units, collecting all our results for the three gauge groups.To relate the bare lattice propagator to the renormalizedcontinuum propagator DR(q; µ) one needs the renormal-ization constant Z3(µ, a):a2D(qa) = Z3(µ, a)DR(q; µ) . (1)We take as our renormalization conditionDR(q)jq2=µ2 = 1µ2(2)at a renormalization scale µ which allows a de-termination of Z3(µ, a). Taking the renormaliza-tion point at µ = 1.94 GeV we nd for SU(3)ZSU(3)3 (µ, a) = 2.51(1), and for SU(2) and SU(4)the ratios ZSU(3)3 (µ, a)/ZSU(2)3 (µ, a) = 1.07(4) andZSU(3)3 (µ, a)/ZSU(4)3 (µ, a) = 0.96(2) respectively..FIG. 2. Volume dependence of the renormalised zero mo-mentum gluon propagator D(0) for SU(2) (crosses), SU(3)(triangles) and SU(4) (stars). The dashed line is the best fitto the form a exp(−V/V0) + c yielding V0 = 0.1 fm4.All the results for various volumes fall rather well ona universal curve demonstrating independence on Nc. Inparticular both in SU(2) and SU(3) they reach a plateauat approximately similar volumes. Since the zero mo-mentum propagator gives a measure of the range overwhich the gauge elds are correlated, these results showthat the gauge elds decorrelate in SU(Nc) at a distanceof  0.8 fm or about 1.7 times the conning correlationlength.In Fig. 3 we plot the results for the renormalized trans-verse gluon propagator for lattice sizes for which we ex-pect volume eects to be negligible, namely 163  32 forSU(3) and SU(4), and 324 for SU(2). Remarkably, theoverall shape of the gluon propagator shows no clear Ncdependence.2FIG. 3. The momentum dependence of the renormalisedtransverse gluon propagator for SU(2) (crosses), SU(3) (tri-angles) and SU(4) (stars) .However, infrared information is suppressed whenshowing q2D(q2). To make any low-momentum dier-ence clearly visible, we plot the SU(2) and SU(4) propa-gators versus the SU(3) propagator for various momenta.If there would be no Nc-dependence the results should fallon the diagonal y = x. Small, but systematic dierencesare visible between SU(3) and SU(2) − −SU(4) wherethe propagators are the largest, i.e. in the infrared, for\physical" momenta smaller than  1 GeV. The SU(3)propagator is slightly larger, which will translate into aslightly smaller gluon pole mass.FIG. 4. The renormalised transverse gluon propagator forSU(2) (crosses) and SU(4) (squares) versus that for SU(3).To investigate more closely the q dependence of thegluon propagator we perform various ts to phenomeno-logical Ansa¨tze. We t the transverse propagator toCornwall’s Ansatz which allows for a dynamically gener-ated gluon mass and is physically motivated. We also tto two parametrizations which appeared recently in theliterature, namely Model A [12] and the mass parametri-sation of ref. [10]:Dmass(q) =Z(q2 + m21)[11 + q4/m42+s[log (1 + q2/m2L)]13/22](3)In Fig. 5 we show the quality of the ts to the threeAnsa¨tze: Cornwall’s Ansatz has three t parameters,Model A four and the mass Ansatz ve.FIG. 5. Fits of the gluon propagator to Cornwall’s Ansatz(solid line) , Model A (dashed line) and the mass parametri-sation (dash-dotted line), in SU(2) (upper), SU(3) (middle)and SU(4) (lower).3In accord with ref. [10] which studied the SU(2) Lan-dau gauge, we nd that the mass Ansatz works well forSU(Nc). This is not so surprising since it has the largestnumber of parameters. We nd no compelling reason toprefer this Ansatz to the other two. Cornwall’s Ansatz isthe only one to allow a smooth analytic continuation to−q2 and we will use it for the determination of the polemass.FIG. 6. Inverse propagator in the infrared, and analyticcontinuation to −q2 for the determination of a pole mass: Up-per graph for SU(2), middle for SU(3) and lower for SU(4).We show Cornwall’s Ansatz (solid line), a linear (dotted line),a quadratic (dashed-dotted line) and a cubic (dashed line)polynomial in q2.The salient features of the infrared behaviour of thegluon propagator in the three gauge groups show up byplotting its inverse as a function of q2. From Fig. 6 itcan be seen that the overall behaviour is very similar forthe three gauge groups although, as shown in Fig. 4, notidentical. To make this observation more quantitative weinvestigate the gluon pole mass by analytic continuationto negative values of q2 as shown in Fig. 6. In additionto Cornwall’s Ansatz we use polynomials of increasingdegree in q2 to test the model dependence of the analyticcontinuation. Taking as a mean the value extracted fromCornwall’s Ansatz and as a crude indication of the sys-tematic error the results of the polynomial t, we ndfor the pole mass (1.85  0.30)pσ, (1.52  0.06)pσ and(1.740.27)pσ in SU(2), SU(3) and SU(4) respectively.A study of the gluon propagator for SU(2) in Landaugauge [10] found that the low momentum behaviour sup-ported a non-zero mass of  (1.48 0.05)pσ. Since onlythe mass parametrisation was used in that study and noanalytic continuation was performed, this value can beregarded as a rough estimate consistent with our polemass.IV. CONCLUSIONSWe have extended a previous lattice study of the gluonpropagator in the Laplacian gauge to SU(Nc). In SU(2),we have made an analysis of the nite-size eects, whichindicates that the Laplacian gauge gluon propagator atzero momentum approaches a non-zero constant value inthe innite volume limit, as in SU(3). In fact, this nitenon-zero value is independent of Nc within our statistics.We nd D(0) = 5.85(5) GeV−2.The overall dependence of the gluon propagator on Ncappears extremely weak, and is barely visible, even forinfrared momenta, even for Nc = 2. For all Nc = 2, 3, 4,the infrared behaviour is consistent with a pole mass inthe range 600− 800 MeV.Our results give further support to the view that Nc =2 is a theory already close to the large Nc limit of QCD, aconclusion reached from the study of the string tension [1]and the glueball mass [13] in SU(Nc).Acknowledgements: The SU(3) 163  32 lattice con-gurations were obtained from the Gauge Connectionarchive [14]. We thank H. Panagopoulos for discussions.[1] B. Lucini and M. Teper, Phys.Rev. D64 (2001) 105019.[2] L. Del Debbio, H. Panagopoulos,P. Rossi and E. Vicari,hep-th/0106185; hep-lat/0110101;hep-th/0111090.[3] R. Kobes, G. Kunstatter and A. Rebhan, Phys. Rev.Lett. 64 (1990) 2992; Nucl. Phys. B355, 1.[4] J. M. Cornwall, Phys. Rev. D 26 (1982) 1453.[5] O. Philipsen, Nucl. Phys b521 (2001) 273; hep-lat/0110114.4[6] C. Alexandrou, Ph. de Forcrand and E. Follana, Phys.Rev. D 63 (2001) 094504.[7] C. Alexandrou, Ph. de Forcrand and E. Follana, hep-lat/0112043.[8] J. C. Vink, Uwe-Jens Wiese, Phys.Lett. B289 (1992) 122;J. C. Vink, Phys. Lett. B321 (1994) 239-245; J. C. Vink,Phys.Rev. D51 (1995) 1292.[9] G. S. Bali and K. Schilling, Phys. Rev. D 47 (1993) 661.[10] K. Langfeld, H. Reinhardt and J. Gattnar, hep-ph/0107141.[11] A. Cucchieri and D. Zwanziger, hep-lat/0012024.[12] D. Leinweber, J. I. Skullerud, A. G. Williams, and C.Parrinello, Phys. Rev. D 60 (1999) 094507; Erratum-ibid.D 61 (2000) 079901.[13] B. Lucini and M Teper, JHEP 0106 (2001) 050.[14] http://qcd.nersc.gov.5

The Laplacian Gauge Gluon Propagator in SU(N_c)

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